Algebraic and geometric structures inside the Birkhoff polytope

TitleAlgebraic and geometric structures inside the Birkhoff polytope
Publication TypeJournal Article
Year of Publication2021
AuthorsRajchel-Mieldzioć G, Korzekwa K, Puchała Z, Życzkowski K

The Birkhoff polytope Bd consisting of all bistochastic matrices of order d assists researchers from
many areas, including combinatorics, statistical physics and quantum information. Its subset Ud
of unistochastic matrices, determined by squared moduli of unitary matrices, is of a particular
importance for quantum theory as classical dynamical systems described by unistochastic transition
matrices can be quantised. In order to investigate the problem of unistochasticity we introduce the
set Ld of bracelet matrices that forms a subset of Bd , but a superset of Ud . We prove that for every
dimension d this set contains the set of factorisable bistochastic matrices Fd and is closed under
matrix multiplication by elements of Fd. Moreover, we prove that both Ld and Fd are star-shaped
with respect to the flat matrix. We also analyse the set of d × d unistochastic matrices arising from
circulant unitary matrices, and show that their spectra lie inside d-hypocycloids on the complex
plane. Finally, applying our results to small dimensions, we fully characterise the set of circulant
unistochastic matrices of order d ≤ 4, and prove that such matrices form a monoid for d = 3.